Find the value of \((243)^{0.16} \times (243)^{0.04}\).
Answer: B
When multiplying powers with the same base, we add the exponents.
The expression becomes \((243)^{0.16 + 0.04} = (243)^{0.20}\).
\(0.20\) as a fraction is \(20/100 = 1/5\).
So we need to find \((243)^{1/5}\), which is the fifth root of 243.
We know that \(3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243\).
Therefore, the value is 3.
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What is 25% of 50% of 200?
Answer: B
First, calculate 50% of 200.
50% of 200 = \(0.5 \times 200 = 100\).
Now, calculate 25% of 100.
25% of 100 = 25.
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The sum of the digits of a 2-digit number is 9. When 27 is added to the number, the digits get reversed. The number is:
Answer: B
Let the ten's digit be x and the unit's digit be y. Number = 10x + y.
Given: x + y = 9.
Given: (10x + y) + 27 = 10y + x (reversed number).
\(9x - 9y = -27\), which simplifies to \(x - y = -3\) or \(y - x = 3\).
We have two equations: x + y = 9 and y - x = 3.
Adding them: 2y = 12, so y = 6.
Substituting y=6 into x+y=9, we get x=3.
The number is 36.
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What is the value of \(587 \times 999\)?
Answer: A
We can write 999 as (1000 - 1).
So, \(587 \times 999 = 587 \times (1000 - 1)\).
= \(587 \times 1000 - 587 \times 1\).
= 587000 - 587 = 586413.
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Find the value of \(0.5 \times 0.05 \times 500\).
Answer: B
We can multiply sequentially.
\(0.5 \times 0.05 = 0.025\).
Now, \(0.025 \times 500 = 25 \times 0.5 = 12.5\).
Alternatively, write as fractions: \((\frac{1}{2}) \times (\frac{5}{100}) \times 500 = (\frac{1}{2}) \times (\frac{1}{20}) \times 500 = \frac{500}{40} = \frac{50}{4} = 12.5\).
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If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?
Answer: B
Let the number be x.
According to the first condition: \(\frac{x-4}{6} = 8\).
\(x-4 = 48\), so \(x = 52\).
Now, according to the second condition, subtract 2 from the number: \(52 - 2 = 50\).
Then divide by 5: \(50 \div 5 = 10\).
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The number 311311311311 is divisible by:
Answer: D
The number is formed by repeating '311' four times.
Divisibility by 3: Sum of digits = (3+1+1) × 4 = 5 × 4 = 20. Since 20 is not divisible by 3, the number is not divisible by 3.
Divisibility by 11: Sum of alternate digits. (1+3+1+3+1+3) - (1+1+1+1) = 12 - 4 = 8. No. Let's do it right to left. (1+1+1+1+1+1) - (3+3+3+3) = 6-12=-6. No. Let's do odd/even positions. (3+1+1+1+1+1)-(1+3+1+3+1+3) = 8-12=-4. It is not divisible by 11.
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The difference between a number and its three-fifths is 50. What is the number?
Answer: C
Let the number be x.
The equation is \(x - \frac{3}{5}x = 50\).
\(\frac{5x-3x}{5} = 50\).
\(\frac{2}{5}x = 50\).
\(x = 50 \times \frac{5}{2} = 25 \times 5 = 125\).
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The sum of all two-digit numbers divisible by 5 is:
Answer: C
The two-digit numbers divisible by 5 are 10, 15, 20, ..., 95. This is an arithmetic progression.
First term (a) = 10. Last term (l) = 95. Common difference (d) = 5.
Number of terms (n) = \((\frac{l-a}{d}) + 1 = (\frac{95-10}{5}) + 1 = 17 + 1 = 18\).
Sum = \(\frac{n}{2}(a+l) = \frac{18}{2}(10+95) = 9 \times 105 = 945\).
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A number whose fifth part increased by 4 is equal to its fourth part diminished by 10. The number is:
Answer: C
Let the number be x.
The equation is \(\frac{x}{5} + 4 = \frac{x}{4} - 10\).
\(4 + 10 = \frac{x}{4} - \frac{x}{5}\).
\(14 = \frac{5x - 4x}{20}\).
\(14 = \frac{x}{20}\).
\(x = 14 \times 20 = 280\).
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